As both a researcher and a genealogy librarian, I don't like to give up on any family history question. I always think that searching church records, doing cluster research and reading newspapers, etc. will yield the answer, if only one will take the time.
And often, I am right when offering these possibilities on finding that missing piece: spelling the names slightly different; switching the first and last name; looking in a neighboring county; the list goes on. But sometimes, I know that the information that is sought will probably never be found.
Do you have an ancestor who was a child left abandoned on porch steps, a church pew or at a train station? The authorities would have sought the mother at the time, so once a search of the records of the abandonment have been searched and local newspapers have been perused, it's unlikely that the identity of the parents will be established. (Although my compulsion to continue the search tells me that if one reads all personal diaries and newspapers for the 25 closest counties and all communities on the connecting rail lines, it may be possible that someone mentioned a pregnant woman....) Or if your ancestor was a woman who married a man that was new in the community, about whom nothing was known, and who disappeared the minute the woman got pregnant and no records were ever found concerning the man.... Well, although one might get lucky in that he really did use his own name, chances are good that he made a practice of loving and leaving, and his real identity may never be discovered.
While I am not advocating completely giving up on the brick wall, I am suggesting setting it aside for a while. New records become available on a regular basis, and you may pick up more experience over several years working on something else. Sometime in the future, you may come across something new. Or you might attend a conference where someone else, a speaker or fellow attendee, could make a break-through suggestion. Or perhaps genetic identification might make such strides that our whole ancestry will be identified with a single drop of blood, and family history research will only give substance to that information.*
But there is always the possibility that the puzzle piece may never be found. This can be hard to accept, especially if you have a wall chart that displays a huge blank for 12.5% of the page. I've seen people frustrated, angry and dispairing over research issues like this, and it very sad because these researchers have exhausted every avenue, made every right move and still come up empty. So the important thing at this point is for us, as researchers, to accept that this may be an unsolvable problem, but not let it ruin our love of family history.
Besides, you never know when you might get lucky....
* Yes, I read science fiction.